Puzzles and mysteries are both forms of deception, but they're different.

Puzzles have one answer. Once you discover that answer you've solved the puzzle. Puzzles rely on one side keeping a secret and the other side finding out the secret.

Mysteries depend on your skill. All the information is right in front of you, and it's up to you to figure out what's going on.

Gregory F. Treverton talks about these differences in his book Reshaping National Intelligence for an Age of Information:

"Cold War intelligence lived in a world where information was scarce; it relied on "secrets" not otherwise available. Its business was those secrets. Now, though, it faces an era of information. Information and its sources are mushrooming, and so are the technologies for moving information rapidly around the globe. Given these circumstances, the business of intelligence is no longer just to provide secrets; rather, its business is to produce high-quality understanding of the world using all sources."

In an article in The New Yorker, Malcolm Gladwell explains these differences as they're related to the Enron scandal, where shifty accounting practices led to the destruction of both the energy company Enron and the accounting firm Arthur Anderson. Mr. Gladwell touches on Enron's Jeffrey Skilling, Watergate, the German secret weapon the V-1 rocket, prostate cancer, and the Cold War:

"...Osama bin Laden’s whereabouts are a puzzle. We can’t find him because we don’t have enough information. The key to the puzzle will probably come from someone close to bin Laden, and until we can find that source bin Laden will remain at large.

The problem of what would happen in Iraq after the toppling of Saddam Hussein was, by contrast, a mystery. It wasn’t a question that had a simple, factual answer. Mysteries require judgments and the assessment of uncertainty, and the hard part is not that we have too little information but that we have too much...

If things go wrong with a puzzle, identifying the culprit is easy: it’s the person who withheld information. Mysteries, though, are a lot murkier: sometimes the information we’ve been given is inadequate, and sometimes we aren’t very smart about making sense of what we’ve been given, and sometimes the question itself cannot be answered. Puzzles come to satisfying conclusions. Mysteries often don’t.

If you sat through the trial of Jeffrey Skilling, you’d think that the Enron scandal was a puzzle. The company, the prosecution said, conducted shady side deals that no one quite understood. Senior executives withheld critical information from investors. Skilling, the architect of the firm’s strategy, was a liar, a thief, and a drunk. We were not told enough—the classic puzzle premise—was the central assumption of the Enron prosecution...

But the prosecutor was wrong. Enron wasn’t really a puzzle. It was a mystery."